Vector being perpendicular to two other vectors

Question

I am having trouble finding a vector that is perpendicular to both $\vec a =-9\hat i+2\hat j+6\hat k$ and $\vec b =-2\hat i+\hat j+\hat k$.

Would it just be $18i + 2j + 6k?$

Answer 1

Find any solution of

$$\begin{cases}-9x+2y+6z&=0,\\-2x+y+z&=0.\end{cases}$$

Set one of the variables arbitrarily and solve for the other two: $$x=4,y=3,z=5.$$

Answer 2

Let $\vec a=-9\hat i+2\hat j+6\hat k$ and $\vec b=-2\hat i+\hat j+\hat k$

Any vector perpendicular to both $\vec a$ and $\vec b$ would be parallel to $\vec a\times \vec b$ because $\vec a\times \vec b$ always yield a vector perpendicular to both the vectors.

Let the required vector be $\vec c$, hence $$\vec c=\lambda(\vec a\times \vec b)$$ $$\vec c=\lambda(-4\hat i-3\hat j-5\hat k)=-4\lambda \hat i-3\lambda \hat j-5\lambda\hat k$$